3.888 \(\int \frac{x^3}{(a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx\)

Optimal. Leaf size=259 \[ \frac{\sqrt [4]{a+b x} (c+d x)^{3/4} \left (77 a^2 d^2-4 b d x (11 a d+9 b c)+54 a b c d+45 b^2 c^2\right )}{96 b^3 d^3}-\frac{\left (77 a^3 d^3+21 a^2 b c d^2+15 a b^2 c^2 d+15 b^3 c^3\right ) \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{64 b^{15/4} d^{13/4}}-\frac{\left (77 a^3 d^3+21 a^2 b c d^2+15 a b^2 c^2 d+15 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{64 b^{15/4} d^{13/4}}+\frac{x^2 \sqrt [4]{a+b x} (c+d x)^{3/4}}{3 b d} \]

[Out]

(x^2*(a + b*x)^(1/4)*(c + d*x)^(3/4))/(3*b*d) + ((a + b*x)^(1/4)*(c + d*x)^(3/4)
*(45*b^2*c^2 + 54*a*b*c*d + 77*a^2*d^2 - 4*b*d*(9*b*c + 11*a*d)*x))/(96*b^3*d^3)
- ((15*b^3*c^3 + 15*a*b^2*c^2*d + 21*a^2*b*c*d^2 + 77*a^3*d^3)*ArcTan[(d^(1/4)*
(a + b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))])/(64*b^(15/4)*d^(13/4)) - ((15*b^3*c
^3 + 15*a*b^2*c^2*d + 21*a^2*b*c*d^2 + 77*a^3*d^3)*ArcTanh[(d^(1/4)*(a + b*x)^(1
/4))/(b^(1/4)*(c + d*x)^(1/4))])/(64*b^(15/4)*d^(13/4))

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Rubi [A]  time = 0.44758, antiderivative size = 259, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{\sqrt [4]{a+b x} (c+d x)^{3/4} \left (77 a^2 d^2-4 b d x (11 a d+9 b c)+54 a b c d+45 b^2 c^2\right )}{96 b^3 d^3}-\frac{\left (77 a^3 d^3+21 a^2 b c d^2+15 a b^2 c^2 d+15 b^3 c^3\right ) \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{64 b^{15/4} d^{13/4}}-\frac{\left (77 a^3 d^3+21 a^2 b c d^2+15 a b^2 c^2 d+15 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{64 b^{15/4} d^{13/4}}+\frac{x^2 \sqrt [4]{a+b x} (c+d x)^{3/4}}{3 b d} \]

Antiderivative was successfully verified.

[In]  Int[x^3/((a + b*x)^(3/4)*(c + d*x)^(1/4)),x]

[Out]

(x^2*(a + b*x)^(1/4)*(c + d*x)^(3/4))/(3*b*d) + ((a + b*x)^(1/4)*(c + d*x)^(3/4)
*(45*b^2*c^2 + 54*a*b*c*d + 77*a^2*d^2 - 4*b*d*(9*b*c + 11*a*d)*x))/(96*b^3*d^3)
- ((15*b^3*c^3 + 15*a*b^2*c^2*d + 21*a^2*b*c*d^2 + 77*a^3*d^3)*ArcTan[(d^(1/4)*
(a + b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))])/(64*b^(15/4)*d^(13/4)) - ((15*b^3*c
^3 + 15*a*b^2*c^2*d + 21*a^2*b*c*d^2 + 77*a^3*d^3)*ArcTanh[(d^(1/4)*(a + b*x)^(1
/4))/(b^(1/4)*(c + d*x)^(1/4))])/(64*b^(15/4)*d^(13/4))

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Rubi in Sympy [A]  time = 38.7167, size = 260, normalized size = 1. \[ \frac{x^{2} \sqrt [4]{a + b x} \left (c + d x\right )^{\frac{3}{4}}}{3 b d} + \frac{\sqrt [4]{a + b x} \left (c + d x\right )^{\frac{3}{4}} \left (\frac{77 a^{2} d^{2}}{16} + \frac{27 a b c d}{8} + \frac{45 b^{2} c^{2}}{16} - \frac{b d x \left (11 a d + 9 b c\right )}{4}\right )}{6 b^{3} d^{3}} - \frac{\left (77 a^{3} d^{3} + 21 a^{2} b c d^{2} + 15 a b^{2} c^{2} d + 15 b^{3} c^{3}\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt [4]{a + b x}}{\sqrt [4]{b} \sqrt [4]{c + d x}} \right )}}{64 b^{\frac{15}{4}} d^{\frac{13}{4}}} - \frac{\left (77 a^{3} d^{3} + 21 a^{2} b c d^{2} + 15 a b^{2} c^{2} d + 15 b^{3} c^{3}\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{d} \sqrt [4]{a + b x}}{\sqrt [4]{b} \sqrt [4]{c + d x}} \right )}}{64 b^{\frac{15}{4}} d^{\frac{13}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(b*x+a)**(3/4)/(d*x+c)**(1/4),x)

[Out]

x**2*(a + b*x)**(1/4)*(c + d*x)**(3/4)/(3*b*d) + (a + b*x)**(1/4)*(c + d*x)**(3/
4)*(77*a**2*d**2/16 + 27*a*b*c*d/8 + 45*b**2*c**2/16 - b*d*x*(11*a*d + 9*b*c)/4)
/(6*b**3*d**3) - (77*a**3*d**3 + 21*a**2*b*c*d**2 + 15*a*b**2*c**2*d + 15*b**3*c
**3)*atan(d**(1/4)*(a + b*x)**(1/4)/(b**(1/4)*(c + d*x)**(1/4)))/(64*b**(15/4)*d
**(13/4)) - (77*a**3*d**3 + 21*a**2*b*c*d**2 + 15*a*b**2*c**2*d + 15*b**3*c**3)*
atanh(d**(1/4)*(a + b*x)**(1/4)/(b**(1/4)*(c + d*x)**(1/4)))/(64*b**(15/4)*d**(1
3/4))

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Mathematica [C]  time = 0.340203, size = 168, normalized size = 0.65 \[ \frac{(c+d x)^{3/4} \left (d (a+b x) \left (77 a^2 d^2+2 a b d (27 c-22 d x)+b^2 \left (45 c^2-36 c d x+32 d^2 x^2\right )\right )-\left (77 a^3 d^3+21 a^2 b c d^2+15 a b^2 c^2 d+15 b^3 c^3\right ) \left (\frac{d (a+b x)}{a d-b c}\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{b (c+d x)}{b c-a d}\right )\right )}{96 b^3 d^4 (a+b x)^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/((a + b*x)^(3/4)*(c + d*x)^(1/4)),x]

[Out]

((c + d*x)^(3/4)*(d*(a + b*x)*(77*a^2*d^2 + 2*a*b*d*(27*c - 22*d*x) + b^2*(45*c^
2 - 36*c*d*x + 32*d^2*x^2)) - (15*b^3*c^3 + 15*a*b^2*c^2*d + 21*a^2*b*c*d^2 + 77
*a^3*d^3)*((d*(a + b*x))/(-(b*c) + a*d))^(3/4)*Hypergeometric2F1[3/4, 3/4, 7/4,
(b*(c + d*x))/(b*c - a*d)]))/(96*b^3*d^4*(a + b*x)^(3/4))

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Maple [F]  time = 0.052, size = 0, normalized size = 0. \[ \int{{x}^{3} \left ( bx+a \right ) ^{-{\frac{3}{4}}}{\frac{1}{\sqrt [4]{dx+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b*x+a)^(3/4)/(d*x+c)^(1/4),x)

[Out]

int(x^3/(b*x+a)^(3/4)/(d*x+c)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{{\left (b x + a\right )}^{\frac{3}{4}}{\left (d x + c\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/((b*x + a)^(3/4)*(d*x + c)^(1/4)),x, algorithm="maxima")

[Out]

integrate(x^3/((b*x + a)^(3/4)*(d*x + c)^(1/4)), x)

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Fricas [A]  time = 0.378718, size = 2253, normalized size = 8.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/((b*x + a)^(3/4)*(d*x + c)^(1/4)),x, algorithm="fricas")

[Out]

1/384*(12*b^3*d^3*((50625*b^12*c^12 + 202500*a*b^11*c^11*d + 587250*a^2*b^10*c^1
0*d^2 + 2092500*a^3*b^9*c^9*d^3 + 4614975*a^4*b^8*c^8*d^4 + 8958600*a^5*b^7*c^7*
d^5 + 18926460*a^6*b^6*c^6*d^6 + 27042120*a^7*b^5*c^5*d^7 + 36722511*a^8*b^4*c^4
*d^8 + 52655988*a^9*b^3*c^3*d^9 + 43080114*a^10*b^2*c^2*d^10 + 38348772*a^11*b*c
*d^11 + 35153041*a^12*d^12)/(b^15*d^13))^(1/4)*arctan((b^4*d^4*x + b^4*c*d^3)*((
50625*b^12*c^12 + 202500*a*b^11*c^11*d + 587250*a^2*b^10*c^10*d^2 + 2092500*a^3*
b^9*c^9*d^3 + 4614975*a^4*b^8*c^8*d^4 + 8958600*a^5*b^7*c^7*d^5 + 18926460*a^6*b
^6*c^6*d^6 + 27042120*a^7*b^5*c^5*d^7 + 36722511*a^8*b^4*c^4*d^8 + 52655988*a^9*
b^3*c^3*d^9 + 43080114*a^10*b^2*c^2*d^10 + 38348772*a^11*b*c*d^11 + 35153041*a^1
2*d^12)/(b^15*d^13))^(1/4)/((15*b^3*c^3 + 15*a*b^2*c^2*d + 21*a^2*b*c*d^2 + 77*a
^3*d^3)*(b*x + a)^(1/4)*(d*x + c)^(3/4) + (d*x + c)*sqrt(((225*b^6*c^6 + 450*a*b
^5*c^5*d + 855*a^2*b^4*c^4*d^2 + 2940*a^3*b^3*c^3*d^3 + 2751*a^4*b^2*c^2*d^4 + 3
234*a^5*b*c*d^5 + 5929*a^6*d^6)*sqrt(b*x + a)*sqrt(d*x + c) + (b^8*d^7*x + b^8*c
*d^6)*sqrt((50625*b^12*c^12 + 202500*a*b^11*c^11*d + 587250*a^2*b^10*c^10*d^2 +
2092500*a^3*b^9*c^9*d^3 + 4614975*a^4*b^8*c^8*d^4 + 8958600*a^5*b^7*c^7*d^5 + 18
926460*a^6*b^6*c^6*d^6 + 27042120*a^7*b^5*c^5*d^7 + 36722511*a^8*b^4*c^4*d^8 + 5
2655988*a^9*b^3*c^3*d^9 + 43080114*a^10*b^2*c^2*d^10 + 38348772*a^11*b*c*d^11 +
35153041*a^12*d^12)/(b^15*d^13)))/(d*x + c)))) - 3*b^3*d^3*((50625*b^12*c^12 + 2
02500*a*b^11*c^11*d + 587250*a^2*b^10*c^10*d^2 + 2092500*a^3*b^9*c^9*d^3 + 46149
75*a^4*b^8*c^8*d^4 + 8958600*a^5*b^7*c^7*d^5 + 18926460*a^6*b^6*c^6*d^6 + 270421
20*a^7*b^5*c^5*d^7 + 36722511*a^8*b^4*c^4*d^8 + 52655988*a^9*b^3*c^3*d^9 + 43080
114*a^10*b^2*c^2*d^10 + 38348772*a^11*b*c*d^11 + 35153041*a^12*d^12)/(b^15*d^13)
)^(1/4)*log(((15*b^3*c^3 + 15*a*b^2*c^2*d + 21*a^2*b*c*d^2 + 77*a^3*d^3)*(b*x +
a)^(1/4)*(d*x + c)^(3/4) + (b^4*d^4*x + b^4*c*d^3)*((50625*b^12*c^12 + 202500*a*
b^11*c^11*d + 587250*a^2*b^10*c^10*d^2 + 2092500*a^3*b^9*c^9*d^3 + 4614975*a^4*b
^8*c^8*d^4 + 8958600*a^5*b^7*c^7*d^5 + 18926460*a^6*b^6*c^6*d^6 + 27042120*a^7*b
^5*c^5*d^7 + 36722511*a^8*b^4*c^4*d^8 + 52655988*a^9*b^3*c^3*d^9 + 43080114*a^10
*b^2*c^2*d^10 + 38348772*a^11*b*c*d^11 + 35153041*a^12*d^12)/(b^15*d^13))^(1/4))
/(d*x + c)) + 3*b^3*d^3*((50625*b^12*c^12 + 202500*a*b^11*c^11*d + 587250*a^2*b^
10*c^10*d^2 + 2092500*a^3*b^9*c^9*d^3 + 4614975*a^4*b^8*c^8*d^4 + 8958600*a^5*b^
7*c^7*d^5 + 18926460*a^6*b^6*c^6*d^6 + 27042120*a^7*b^5*c^5*d^7 + 36722511*a^8*b
^4*c^4*d^8 + 52655988*a^9*b^3*c^3*d^9 + 43080114*a^10*b^2*c^2*d^10 + 38348772*a^
11*b*c*d^11 + 35153041*a^12*d^12)/(b^15*d^13))^(1/4)*log(((15*b^3*c^3 + 15*a*b^2
*c^2*d + 21*a^2*b*c*d^2 + 77*a^3*d^3)*(b*x + a)^(1/4)*(d*x + c)^(3/4) - (b^4*d^4
*x + b^4*c*d^3)*((50625*b^12*c^12 + 202500*a*b^11*c^11*d + 587250*a^2*b^10*c^10*
d^2 + 2092500*a^3*b^9*c^9*d^3 + 4614975*a^4*b^8*c^8*d^4 + 8958600*a^5*b^7*c^7*d^
5 + 18926460*a^6*b^6*c^6*d^6 + 27042120*a^7*b^5*c^5*d^7 + 36722511*a^8*b^4*c^4*d
^8 + 52655988*a^9*b^3*c^3*d^9 + 43080114*a^10*b^2*c^2*d^10 + 38348772*a^11*b*c*d
^11 + 35153041*a^12*d^12)/(b^15*d^13))^(1/4))/(d*x + c)) + 4*(32*b^2*d^2*x^2 + 4
5*b^2*c^2 + 54*a*b*c*d + 77*a^2*d^2 - 4*(9*b^2*c*d + 11*a*b*d^2)*x)*(b*x + a)^(1
/4)*(d*x + c)^(3/4))/(b^3*d^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\left (a + b x\right )^{\frac{3}{4}} \sqrt [4]{c + d x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(b*x+a)**(3/4)/(d*x+c)**(1/4),x)

[Out]

Integral(x**3/((a + b*x)**(3/4)*(c + d*x)**(1/4)), x)

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/((b*x + a)^(3/4)*(d*x + c)^(1/4)),x, algorithm="giac")

[Out]

Timed out